Once a conjecture is formally proven true it is elevated to the status of theorem and may be used afterwards without risk in the construction of other formal mathematical proofs.

In scientific philosophy, Karl Popper pioneered the use of conjecture to indicate a statement which is presumed to be real, true, or genuine, mostly based on inconclusive grounds, in contrast with a hypothesis (hence theory, axiom, principle), which is a testable statement based on accepted grounds.

Until recently, the most famous conjecture was the mis-named Fermat's last theorem, mis-named because although Fermat claimed to have found a clever proof of it, none could be found among his notes after his death.

In its original form, now known as the weak Goldbach conjecture, it was put forward by the Prussian amateur mathematician and historian Christian Goldbach (1690-1764) in a letter dated Jun. 7, 1742, to Leonhard Euler.

In this guise it says that every whole number greater than 5 is the sum of three prime numbers.

Euler restated this, in an equivalent form, as what is now called the strong Goldbach conjecture or, simply, the Goldbach conjecture: every even number greater than 2 is the sum of two primes.

Ramaré95] (Goldbach's conjecture suggests two) and in 1966 Chen proved every sufficiently large even integers is the sum of a prime plus a number with no more than two prime factors (a P

Goldbach conjecture also showed that every even number is the difference between a prime and a P

Hardy and Wright give an argument for this conjecture in their well known footnote [HW79, p15] which goes roughly as follows.

primes.utm.edu /notes/conjectures (566 words)

Goldbach's Conjecture(Site not responding. Last check: 2007-10-31)

In his famous letter to Leonhard Euler dated June 7th 1742, Christian Goldbach first conjectures that every number that is a sum of two primes can be written as a sum of "as many primes as one wants".

On the margin of his letter, he then states his famous conjecture that every number is a sum of three primes:

The ternary conjecture has been proved under the assumption of the truth of the generalized Riemann hypothesis and remains unproved unconditionally for only a finite (but yet not computationally coverable) set of numbers.

www.mscs.dal.ca /~joerg/res/g-en.html (390 words)

The Poincare conjecture(Site not responding. Last check: 2007-10-31)

The Poincare Conjecture is essentially the first conjecture ever made in topology; it asserts that a 3-dimensional manifold is the same as the 3-dimensional sphere precisely when a certain algebraic condition is satisfied.

The conjecture was formulated by Poincare around the turn of the 20th century.

The Generalized Poincare Conjecture states that for every n, an n-dimensional manifold homotopy equivalent to the n-sphere is homeomorphic to the n-sphere.

One simple and useful result from the Jacobian conjecture is that by a simple reduction argument we know if there is a counter example to the Jacobian conjecture, then there must be a counter example $f, g$ with the degrees of $f, g$ non-divisible by each other.

Abhyankar was one of the main movers of this conjecture and motivated research on the subject.

This conjecture could be understood by anyone with a background in Calculus and hence it was studied by mathematicians in many disciplines, especially Algebra, Analysis, and Complex Geometry.

That conjecture dates back to 1955, when it was published in Japanese as a research problem by the late Yutaka Taniyama.

The key element appears to be a problem termed the ABCconjecture, which was formulated in the mid-1980s by Joseph Oesterle of the University of Paris VI and David W. Masser of the Mathematics Institute of the University of Basel in Switzerland.

The 3x+1 conjecture [1], [2, problem E16] asserts that starting from any positive integer n the repeated iteration of T(x) eventually produces the integer 1, after which the iterates will alternate between the integers 1 and 2.

In order to test the 3x+1 conjecture, in 1996 we wrote a computer program (in the programming language C), which computed the trajectories of all initial values of n smaller that a given limit and having a stopping time known to be larger than 40 [3].

Since no counter-example was found, the 3x+1 conjecture is probably true for all positive integers not larger than this verification limit.

For 100 years mathematicians have been trying to prove a conjecture that was first proposed by Henri Poincar¿ relating to an object known as the three-dimensional sphere, or 3-sphere.

Here I focus on Poincar¿ himself and the early years of his conjecture, in particular the astonishing results that proved higher-dimensional versions of the conjecture in the latter half of the twentieth century.

Smale heard about the Poincar¿ conjecture in 1955, while he was a graduate student at the University of Michigan in Ann Arbor.

A wide array of sophisticated mathematical techniques could be used in the attempt to prove the conjecture true (and the majority of mathematicians competent to judge seem to believe that it likely is true).

But, I said, I did recall another contest related to Fermat's last Theorem, which would also be difficult to prove true, but which might be shown to be false with some adding and multiplying (of numbers that are a bit beyond the typical kindergarden range).

The mathematicians' honeycomb conjecture therefore concerns a two-dimensional pattern-as if bees were creating a grid for laying out tiles to cover an infinitely wide bathroom floor.

Fejes Tóth proved the honeycomb conjecture for the special case of filling the plane with any mixture of straight-sided polygons.

In recent years, Morgan has refocused attention on the honeycomb conjecture and related questions, such as the most economical way of packaging a pair of identical volumes as double bubbles (SN: 8/12/95, p.

www.sciencenews.org /sn_arc99/7_24_99/bob2.htm (1749 words)

Pitt math professor took best shot at cannonball conjecture(Site not responding. Last check: 2007-10-31)

But for four centuries, mathematicians had been unable to prove famed astronomer and mathematician Johannes Kepler's 1611 conjecture that the pyramid is the best way to stack cannonballs.

The proof is about 300 pages long -- not counting 40,000 lines of computer code and three billion bytes of data necessary to solve the puzzle that left mathematicians scratching their scalps for centuries.

Since proving the Kepler conjecture, he also solved the honeycomb conjecture by proving that the hexagon-shaped cells in honeycombs maximize area and minimize the amount of beeswax.